On recurrence and ergodicity for geodesic flows on noncompact periodic polygonal surfaces

Abstract

We study the recurrence and ergodicity for the billiard on noncompact polygonal surfaces with a free, cocompact action of $\mathbb{Z}$ or ${\mathbb{Z}}^2$. In the $\mathbb{Z}$-periodic case, we establish criteria for recurrence. In the more difficult ${\mathbb{Z}}^2$-periodic case, we establish some general results. For a particular family of ${\mathbb{Z}}^2$-periodic polygonal surfaces, known in the physics literature as the wind-tree model, assuming certain restrictions of geometric nature, we obtain the ergodic decomposition of directional billiard dynamics for a dense, countable set of directions. This is a consequence of our results on the ergodicity of $\ZZ$-valued cocycles over irrational rotations.

48 pages, 12 figures

References in corpus (1)

• Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces

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